A115415 Real part of (n + i)^n, with i=sqrt(-1).
1, 1, 3, 18, 161, 1900, 27755, 482552, 9722113, 222612624, 5707904499, 161981127968, 5039646554593, 170561613679808, 6237995487261915, 245159013138710400, 10303367499652761601, 461102348510408544512, 21891769059478538933603, 1098983344602124698522112
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..386
Programs
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Mathematica
Table[ Re[(n + I)^n], {n, 0, 17}] (* Robert G. Wilson v, Jan 23 2006 *) -
PARI
a(n) = real((n + I)^n); \\ Michel Marcus, Apr 11 2018
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Python
from math import comb def A115415(n): return sum(comb(n,j)*n**(n-j)*(-1 if j&2 else 1) for j in range(0,n+1,2)) # Chai Wah Wu, Feb 15 2024
Formula
a(n) = n! * [x^n] exp(n*x)*cos(x). - Ilya Gutkovskiy, Apr 10 2018
a(n) ~ cos(1) * n^n. - Vaclav Kotesovec, Jun 08 2019
a(n) = Sum_{j=0..floor(n/2)} binomial(n,2j)*n^(n-2j)*(-1)^j. - Chai Wah Wu, Feb 15 2024
a(n) = (1/2)*((n + i)^n + (n - i)^n) where i is the imaginary unit. - Gerry Martens, Dec 30 2024
Extensions
More terms from Robert G. Wilson v, Jan 23 2006